Linear Algebra of Eigen s Quasispecies Model

Transcription

1 Linear Algebra of Eigen s Quasispecies Model Artem Novozhilov Department of Mathematics North Dakota State University Midwest Mathematical Biology Conference, University of Wisconsin La Crosse May 17, 2014 Artem Novozhilov (NDSU) May 17, / 29

2 Collaborators: Yuri S. Semenov Moscow State University of Railway Engineering, Moscow, Russia Alexander S. Bratus Lomonosov Moscow State University Moscow, Russia Artem Novozhilov (NDSU) May 17, / 29

3 Outline: Motivation and historic outline Mathematical problem Known results New results Artem Novozhilov (NDSU) May 17, / 29

4 M. Eigen, Naturwisenschaften, 58(10), 1971: Artem Novozhilov (NDSU) May 17, / 29

5 Quasispecies theory: M. Eigen, Naturwisenschaften, 58(10), 1971: M. Eigen, P. Schuster, The Hypercycle, Springer, 1979 M. Eigen, J. McCaskill, P. Schuster, J Phys Chem, 92(24), 1988: Manfred Eigen, born 1927 Artem Novozhilov (NDSU) May 17, / 29

6 Mathematical side of the story: Ref: M. Eigen, J. McCaskill, P. Schuster, J Phys Chem, 92(24), 1988: Artem Novozhilov (NDSU) May 17, / 29

7 Model statement: Consider a population of sequences of fixed length N composed of two-letter alphabet, say, {0,1}, therefore 2 N different sequences. The population is subject to two evolutionary forces. First evolutionary force is selection, which is included in the system through the Malthusian fitness, defined here for simplicity as m(particular sequence σ) = m(h σ ), where H σ is the Hamming norm of this sequence, i.e., number of 1s in sequence σ. In this way we do not distinguish between sequences with the same number of 1s and hence reduce the dimensionality of the problem from 2 N 2 N to (N +1) (N +1). Hence, we consider at this point only permutation invariant fitness landscapes M = diag(m 0,...,m N ) or m = (m 0,...,m N ). Artem Novozhilov (NDSU) May 17, / 29

8 Model statement: The second evolutionary force is mutation. In particular, assuming N +1 classes of sequences, we have that the mutations µ ij (i.e., the mutation rate from class j to class i) can be described by the matrix N N N N 1 N M = (µ ij ) = µq = 0 0 N 2 N , N N N where µ is the mutation rate per site per sequence per replication event. Artem Novozhilov (NDSU) May 17, / 29

9 Model statement: Let p(t) denote the vector of frequencies of different classes of sequences, then, assuming uncoupled reproduction and mutation events, we arrive at where is the mean population fitness. ṗ(t) = (M +µq)p(t) m(t)p(t), m(t) = m p(t) = N m i p i (t) This model is often called a paramuse of Crow Kimura quasispecies model with permutation invariant fitness landscape. Ref: Baake and Gabriel, Annual Reviews of Computational Physics VII, 1999: Ref: Crow and Kimura, An introduction to population genetics theory, 1970 i=0 Artem Novozhilov (NDSU) May 17, / 29

10 Elementary results: The asymptotic behavior of the quasispecies model is determined by the equilibrium p = lim t p(t), which solves the eigenvalue problem where (M +µq)p = mp, m = m p. By Perron Frobenius theorem it follows that there is a unique positive solution p > 0, which is the right eigenvector of M +µq corresponding to the simple real dominant eigenvalue λ = m. This vector p was called by Eigen the quasispecies. It is globally stable for the quasispecies system. We are mostly interested in properties of m and p depending on the fitness landscape M and mutation rate µ, therefore, we use the notation m = m(µ) and p = p(µ) for the mean fitness and equilibrium distribution. Artem Novozhilov (NDSU) May 17, / 29

11 Known results: Thompson and McBride, Math Biosciences, 21: , 1974 The quasispecies model is essentially linear. Rumschitzki, J of Math Biol, 24: , 1987 For zero epistasis fitness landscape the spectral properties of the Eigen evolutionary matrices can be inferred with the representation of M and M using tensor products. Swetina and Schuster, Bioph Chem, 16: , 1982 Numerical analysis of single peaked fitness landscape yields the error threshold. Artem Novozhilov (NDSU) May 17, / 29

12 Known results: The error threshold Ref: Swetina and Schuster, Bioph Chem, 16: , 1982 Consider the single peaked fitness landscape M = diag(m 0,0,...,0), m 0 > 0. Artem Novozhilov (NDSU) May 17, / 29

13 Known results: The error threshold Consider the single peaked fitness landscape M = diag(m 0,0,...,0), m 0 > p µ Artem Novozhilov (NDSU) May 17, / 29

14 Known results: Statistical Physics Leuthäusser, I. (1986). An exact correspondence between Eigen s evolution model and a two-dimensional Ising system. The Journal of Chemical Physics, 84(3), Tarazona, P. (1992). Error thresholds for molecular quasispecies as phase transitions: From simple landscapes to spin-glass models. Physical Review A, 45(8), Baake, E., Baake, M., & Wagner, H. (1997). Ising quantum chain is equivalent to a model of biological evolution. Physical Review Letters, 78(3), Galluccio, S. (1997). Exact solution of the quasispecies model in a sharply peaked fitness landscape. Physical Review E, 56(4), Baake, E., & Wagner, H. (2001). Mutation-selection models solved exactly with methods of statistical mechanics. Genetical Research, 78(01), Saakian, D. B., & Hu, C. K. (2006). Exact solution of the Eigen model with general fitness functions and degradation rates. Proceedings of the National Academy of Sciences of the USA, 103(13), Artem Novozhilov (NDSU) May 17, / 29

15 Known results: Maximum principle Hermisson, J., Redner, O., Wagner, H., & Baake, E. (2002). Mutation-selection balance: Ancestry, load, and maximum principle. Theoretical Population Biology, 62(1), Baake, E., & Georgii, H. O. (2007). Mutation, selection, and ancestry in branching models: a variational approach. Journal of Mathematical Biology, 54(2), Assume that m i = Nr i = Nr(x i ), x i = i N [0,1] and define g(x) = µ ( 1 2 x(1 x) ). Then the mean fitness m(µ) = Nr is given by ( ) r r = sup r(x) g(x). x [0,1] r(x) may have only finite number of discontinuities and be either left or right continuous at every point. Artem Novozhilov (NDSU) May 17, / 29

16 Main idea: We consider the eigenvalue problem (M +µq)p = mp, m = m p, where p = p(µ), m = m(µ) with a fixed fitness landscape m. We claim that this problem simplifies in the coordinates of the basis composed of the eigenvectors of the matrix Q = Q N. Artem Novozhilov (NDSU) May 17, / 29

17 Proposition: For the matrix Q = Q N : 1. The eigenvalues of Q N are simple (all have algebraic multiplicities one) and given by q k = 2k, k = 0,...,N. 2. Let v k = (c 0k,...,c Nk ) be the right eigenvector of Q N corresponding to q k and normalized such that c 0k = 1, C = C N = (c ik ) (N+1) (N+1) be the matrix composed of v k (v k is the k-th column of C N). Then the generating function for the elements of the k-th column has the form P k (t) = N c ik t i = (1 t) k (1+t) N k, k = 0,...,N. i=0 3. C 2 = 2 N I, where I is the identity matrix, or, equivalently, C 1 = 2 N C norm of C is C 1 = max 0 k N N c ik = 2 N. i=0 Artem Novozhilov (NDSU) May 17, / 29

18 Behavior for µ : Let x = C 1 p and ˆx = 2 N (1,0,...,0). Then x ˆx 1 1 µ M 1, and Therefore, x (µ) 1 p(µ) ˆp = 2 N (( N 0 2N 2µ (2 N+1 +1) M 1. ) ( )) N,...,. N Artem Novozhilov (NDSU) May 17, / 29

19 Parametric solution: Consider fitness landscape such that m j > 0 for some j, and all the rest m i = 0 for i j. Then x k (s) = 2 N c kj 1+ks, p i (s) = 2 N N k=0 m(s) = m j p j (s), µ = s 2 m(s). c ik c kj 1+ks, Artem Novozhilov (NDSU) May 17, / 29

20 Example 1: Let N = 2A and m = (0,...,0,N,0,...,0), where N is exactly on A-th position. Consider a scaled fitness landscape Nr = m and r(µ) = m(µ)/n. Then, using the properties of the mutation matrix Q and the parametric formulas from the previous slide, it can be proved that r r = lim N r(µ) = µ 2 +1 µ. 1 N N 200 r Μ r Μ 0 3 Μ 0 3 Μ Artem Novozhilov (NDSU) May 17, / 29

21 Example 1: Let N = 2A and m = (0,...,0,N,0,...,0), where N is exactly on A-th position. Consider a scaled fitness landscape Nr = m and r(µ) = m(µ)/n. Denoting r = µ 2 +1 µ we can prove that lim N p A±k(µ) = r ( ) k 1 r, k = 0,...,A. 1+r p p µ µ Artem Novozhilov (NDSU) May 17, / 29

22 Example 2: Single peaked landscape Let m = (N,0,...,0). Consider a scaled fitness landscape Nr = m and r(µ) = m(µ)/n. Then, using the properties of the mutation matrix Q and the parametric formulas we can prove that if µ < 1 and p j = 0 for all j if µ 1. r = lim p 0 = 1 µ, N lim p i = (1 µ)µ i, i 1. N Artem Novozhilov (NDSU) May 17, / 29

23 Example 2: Single peaked landscape Comparison with numerical computations for N = 100. r = lim N p 0 = 1 µ, lim p i = (1 µ)µ i, i 1 N r(µ) p µ µ Artem Novozhilov (NDSU) May 17, / 29

24 Epsilon stabilization: Definition: Dominant eigenvalue m(µ) admits epsilon stabilization if for a small enough ε > 0 there exists constant m ε and µ ε such that for all µ > µ ε we have m(µ) m ǫ < ε, m (µ) < ε. We know that epsilon stabilization is a sure event, due to the previous. The question is how to determine µ ε given ε > 0. Artem Novozhilov (NDSU) May 17, / 29

25 Epsilon stabilization: Let m = (m 0,m 1,...,m N ) such that m 0 > m 1 m 2... m N. Then we have µ = m 0 m 1, (Classical Result) N ( ) µ ε = (m 0 m 1 ) 1 1 2(m 0 1), (m 0 m 1 )N N ( ) (Our result 1) N if δ = 2 N (m k 1) < ε, k k=0 m(µ ) = m min +2µ. (Our result 2) Artem Novozhilov (NDSU) May 17, / 29

26 p Example: m Μ Μ Μ Figure : Error threshold in the quasispecies model with the single picked fitness landscape Estimates from the previous slide: µ 1 = 0.633, µ 2 = 0.644, µ 3 = Artem Novozhilov (NDSU) May 17, / 29

27 Example: p 0.15 m Μ Μ Μ Figure : Error threshold in the quasispecies model with m k = k α log(1 s) Estimates from the previous slide: µ 1 = 0.09, µ 2 = 0.19, µ 3 = 0.33 Artem Novozhilov (NDSU) May 17, / 29

28 Acknowledgements: Startup grant from Department of Mathematics, NDSU ND EPSCoR and NSF grant # EPS Artem Novozhilov (NDSU) May 17, / 29

29 Thank you for your attention! site: novozhil/ References: Bratus, A. S., Novozhilov, A. S., & Semenov, Y. S. (2013). Linear algebra of the permutation invariant Crow Kimura model of prebiotic evolution. arxiv: Semenov, Y. S., Bratus, A. S., & Novozhilov, A. S. (2014). On the behavior of the leading eigenvalue of the Eigen evolutionary matrices, in preparation. Artem Novozhilov (NDSU) May 17, / 29